Mechanics of Materials (MindTap Course List)
Mechanics of Materials (MindTap Course List)
9th Edition
ISBN: 9781337093347
Author: Barry J. Goodno, James M. Gere
Publisher: Cengage Learning
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Chapter 6, Problem 6.2.13P

A simply supported wooden I-beam with a 12-ft span supports a distributed load of intensity q = 90 lb/ft over its length (see figure part a). The beam is constructed with a web of Douglas-fir plywood and flanges of pine glued to the web, as shown in the figure part b. The plywood is 3/8 in. thick: the flanges are 2 in, × 2 in, (actual size). The modulus of elasticity for the plywood is 1,600,000 psi and for the pine is 1,200,000 psL

  1. Calculate the maximum bending stresses in the pine flanges and in the plywood web.

  • What is q, if allowable stresses are 1600 psi in the flanges and 1200 psi in the web?
  •   Chapter 6, Problem 6.2.13P, A simply supported wooden I-beam with a 12-ft span supports a distributed load of intensity q = 90 , example  1

      Chapter 6, Problem 6.2.13P, A simply supported wooden I-beam with a 12-ft span supports a distributed load of intensity q = 90 , example  2

    a.

    Expert Solution
    Check Mark
    To determine

    The bending stress that is maximum in the plywood web and pine flanges

    Answer to Problem 6.2.13P

    Bending Stress Maximum for plywood =1130.629psi

    Bending Stress Maximum for pine =969.11psi

    Explanation of Solution

    Given:

    The given figure

      Mechanics of Materials (MindTap Course List), Chapter 6, Problem 6.2.13P , additional homework tip  1

    The I beam that is wooden has the span of 12 ft over its length supporting a distributed intensity load q=90lb/ft. The beam is made of Douglas fir plywood web and pine flanges with the plywood having a thickness of 3/8 in and that of flanges 2in.*2in. The elasticity modulus of the plywood is 1,600,000psi and for pine it is 1,200,000psi.

    Concept Used:

    Bending Stress Maximum for plywood,

      σpw=Mmax( h 1 2)(E pw)EpwI1+EpI2

    Where,

      MMax=Maximum bending momentI1=Plywood Inertia momentI2=Pine Inertia momentEPW=Plywood modulus ElasticityEP=Pine modulus Elasticityh1=height

    Calculation:

    Bending moment maximum is given as

      Mmax=qL28

    Substituting the values we have,

      Mmax=90*1228

      =90* 1228=90*1448=129608=1620lb.ft

    Plywood moment of inertia is given as,

      I1=t*h1312

      =( 3 8 )*7312=( 3 8 )*34312=128.62512=10.719in4

    Pine moment of inertia is given as,

      I2=2(ba312+ba( h 1 +a 2 )2+(bt) ( h 2 a)312)+(bt)(h2a)( h 1 2 h 2 a2)2

      =2(2* ( 1 2 )312+2*( 1 2 )2( 7+ 1 2 2 )2+( 2 3 8 ) ( 2 1 2 )312)+(238)(212)(72 ( 2 3 8 )2)2

      =2(2* ( 1 2 )312+2*(14)( 3.75)2+0.457)+(138)(32)(72 ( 13 8 )2)2

      =2(0.02083+14.0625+0.457+18.4336)

      =65.948in4

      EpwI1+EpI2=(1600000)(10.718)+(1200000)(65.948)

      =(1600000)(10.718)+(1200000)(65.948)=17148800000+79137600000=96286217in4

    Bending Stress Maximum for plywood,

      σpw=Mmax( h 1 2)(E pw)EpwI1+EpI2

      =( 1620*12)( 7 2 )( 1.6* 10 6 )96286217=( 19440)( 7 2 )( 1.6* 10 6 )96286217=1130.629psi

    Bending Stress Maximum for pine,

      σpw=Mmax( h 1 2+a)(EP)EpwI1+EpI2

      =(1620*12)( 7 2 + 1 2 )( 1200000)96286217=(19440)(4)( 1200000)96286217=969.11psi

    Conclusion:

    Thus, the bending stress that is maximum in the plywood web and pine flanges is given by equating the maximum bending movement

    b.

    Expert Solution
    Check Mark
    To determine

    The qmax with 1600psi maximum stress in flanges and for web being 1200psi.

    Answer to Problem 6.2.13P

      qmax=95.5lb.ft

    Explanation of Solution

    Given:

    The given figure:

      Mechanics of Materials (MindTap Course List), Chapter 6, Problem 6.2.13P , additional homework tip  2

    The I beam that is wooden has the span of 12 ft over its length supporting a distributed intensity load q=90lb/ft. The beam is made of Douglas fir plywood web and pine flanges with the plywood having a thickness of 3/8 in and that of flanges 2in.*2in. The elasticity modulus of the plywood is 1,600,000psi and for pine it is 1,200,000psi.

    Concept Used:

    Web maximum stress,

      σpw=Mpw( h 1 2)(E pw)EpwI1+EpI2

    Where,

      Mpw=Maximum allowable stressI1=Plywood Inertia momentI2=Pine Inertia momentEPW=Plywood modulus ElasticityEP=Pine modulus Elasticityh1=height

    Calculation:

    Maximum stress Maximum for Web,

      σpw=Mpw( h 1 2)(E pw)EpwI1+EpI2

    Substituting the values we have,

      1200=Mpw(72)(1600000)96286217

      1200*96286217=Mpw(72)(1600000)Mpw=1200*96286217( 7 2 )( 1600000)

      =1200*96286217( 7 2 )( 1600000)=72214.663( 7 2 )=20632.76lb.in1719.397lb.ft

    Maximum stress Maximum for flange,

      σpine=Mp( h 1 2+a)(Ep)EpwI1+EpI2

    Substituting the values we have,

      1600=Mp( 7 2 + 1 2 )( 1200000)962862171600*96286217=Mp(4)(1200000)Mp=1600*96286217(4)( 1200000)

      =32095.406lb.in2674.62lb.ft

    Bending moment that is maximum,

      Mall=min(Mpw,Mp)

      1719.40lb.ft

    Maximum uniform load distributed is given as,

      Mall=qmaxL28qmax=8MallL2

      =8*1719.397 122=8*1719.397144=95.522lb.ft

    Conclusion:

    Thus, the qmax with 1600psi maximum stress in flanges and for web being 1200psi is given by the equation of maximum load.

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